18.311 — MIT (Spring 2014)
Rodolfo R. Rosales
March 3, 2014.
Problem Set # 03. Due: Fri. March 14.
IMPORTANT:
— Turn in the regular and the special problems stapled in two SEPARATE packages.
— Print your name in each page of your answers.
— In page one of each package print the names of the other members of your group.
Update: fixed a problem with problem 1.4 below.
Contents
1 Regular Problems.
1.1 Statement: Haberman problem 61.03 . . . . . . .
Formula for car’s acceleration. . . . . . . . . .
1.2 Statement: Haberman problem 63.06 . . . . . . .
Formula for car acceleration. . . . . . . . . .
1.3 Statement: Haberman problem 67.01 . . . . . . .
The simplest velocity-density relationship. . .
1.4 Statement: Haberman problem 71.02 . . . . . . .
Density wave velocity relative to a moving car.
1.5 Problem 6302a. State laws and road capacity. . .
1.6 Statement: Linear 1st order PDE # 05. . . . . . .
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2 Special Problems.
2.1 Statement: TFPa14. Solve using dimensionless variables . . . . . . . . . . . . . . .
Write traffic flow problem in a-dimensional variables, and solve. . . . . . . . . . .
2.2 Statement: TFPa19. Linear first order equation and characteristics . . . . . . . . .
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1
Regular Problems.
1.1
Statement: Haberman problem 61.03.
Assume that a velocity field, u = u(x, t) exists. Show that the acceleration of an individual car is
given by
∂u
∂u
+u .
(1.1)
∂t
∂x
1.2
Statement: Haberman problem 63.06.
Assume that u = u(ρ). If α denotes the car’s acceleration, show that
α = −ρ
du ∂u
.
dρ ∂x
(1.2)
Is the minus sign here reasonable?
1.3
Statement: Haberman problem 67.01.
Suppose that
u = umax 1 −
ρ
ρmax
.
(1.3)
(a) What is the density wave velocity if the traffic density is nearly ρ0 everywhere.
(b) Show that this density wave velocity is less than the car’s velocity.
1.4
Statement: Haberman problem 71.02.
Consider the flow-density relationship used by H. Greenberg 1 to match Lincoln Tunnel data
q = a ρ ln
ρmax
,
ρ
where a > 0 is a constant. Show that the density wave velocity relative to a moving car is the same
constant no matter what the density.
1
Greenberg H., An Analysis of Traffic Flow, Operations Research, 7:79–85, (1959).
2
1.5
Problem 6302a. State laws and road capacity.
If cars obey (exactly) the typical state law 2 on following distances: (1) What is the road capacity?
(2) At what density and velocity does this maximum occur? (3) Will increasing the speed limit
increase the road’s capacity? (4) Draw a plot of the flux q versus the density ρ. (5) Draw a plot of
the flux q versus the velocity u.
The relationship of the car velocity u to the car density ρ prescribed by the state laws has the form
�
u = min uM , V
1
−1
ρL
�
.
(1.4)
Here L is the car length, V is the “trigger” velocity in the law,3 and uM is the speed limit. Assume
that L = 16 ft = (1/330) mile, V = 10 m.p.h., and uM = 50 m.p.h. (80 k.p.h.).
1.6
Statement: Linear 1st order PDE # 05.
Consider the problem
x ux + (x + y) uy = 1,
with
u(1, y) = y for 0 < y < 1.
(1.5)
Question 1. Write the solution u = u(x, y) in the region where it is uniquely determined.
Question 2. Describe the region in the plane where the solution to (1.5) is uniquely determined.
Question 3. Write all the functions u = u(x, y) that satisfy (1.5) on x > 0 and −∞ < y < ∞.
Question 4. Write all the functions u = u(x, y) that satisfy the pde in (1.5) for x < 0.
Question 5 (challenge/optional). What happens along x = 0? Can you produce solutions to the
pde that are continuous in the “punctured” plane (plane minus the origin)?
2
Special Problems.
2.1
Statement: TFPa14. Solve using dimensionless variables.
Consider the traffic flow equation
ρt + qx = 0,
with the quadratic flow q =
4 qm
ρ2j
ρ (ρj − ρ), where qm = road capacity and ρj = jamming density.
Assume that either
2
3
(2.6)
See equation (1.4).
As in: Keep the distance to the next car at one car length L per every V in speed.
3
⎧
⎪
⎨
ρ0
(a) ρ(x, 0) =
⎪
⎩
ρ0
1 + (x/d)2
for x ≤ 0
for x ≥ 0
or
⎧
⎪
⎨
(b) ρ(x, 0) = ⎪
⎩
ρ0
ρ0
1 + (x/d)
for x ≤ 0
for x ≥ 0
where 0 < ρ0 < ρj and d > 0 are constants. Then
1. Write everything using dimensionless variables.
2. Write the solution to (a) and (b) above using characteristics.
3. Show that there are no characteristic crossings. Thus (2) gives the complete solution.
4. In the case of (b), solve for ξ = ξ(x, t), and write the solution explicitly.
Note: in (4) ξ denotes the label for the characteristics, defined as follows: x = ξ at t = 0.
2.2 Statement: TFPa19.
Linear first order equation and characteristics.
Find the solution to
x ψx − y ψy + ψ = 2 x ,
with ψ = 0 on the curve Γ given by y = 1.
Draw the characteristics which touch Γ. Where is the solution defined?
THE END.
4
(2.7)
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18.311 Principles of Applied Mathematics
Spring 2014
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